There are many examples in the prior art of photonic devices which exploit nonlinear optical properties of materials to convert input light with at least one frequency into generated light with at least one frequency different from the input frequency or frequencies. In one type of device, light of a first frequency (f1) passes through a material with a non-zero second order nonlinear susceptibility, and some or all of the input light is converted into generated light with a second frequency (f2), where f2=2f1. This process is commonly referred to as “second harmonic generation” (SHG) and sometimes as “frequency doubling”. In another type of device, input light including light with two different frequencies passes through a material with a non-zero second order nonlinear susceptibility, and some or all of the input light may be converted into generated light with a third frequency. This process is commonly referred to as sum frequency generation (SFG) or difference frequency generation (DFG). SHG, SFG and DFG are all examples of nonlinear frequency conversion (NLFC) processes.
A wide range of materials with large second order nonlinear susceptibilities are known to exist. Examples include lithium niobate (LiNbO3), potassium titanyl phosphate (KTiOPO4), lithium triborate (LiB3O5), β-barium borate (β-BaB2O4) and potassium dihydrogen phosphate (KH2PO4), among many others. A component which can exhibit NLFC can be fabricated from these materials and is referred to here as an NLFC component.
In photonic devices which use NLFC components, it is often important that the NLFC process occurs with high efficiency. The efficiency of NLFC is the ratio of a power of generated light divided by a power of the input light. In the case of SHG the efficiency of SHG is the ratio of the power of frequency-doubled light divided by the power of the input light.
The efficiency of NLFC can depend strongly on the properties of the input beam of light as it passes through an NLFC component. In many examples of NLFC the input beam is focussed in at least one plane of the beam (a plane of the beam is any plane within which the propagation direction of the beam lies). This is illustrated in FIG. 1 which is an illustration of a focussed input beam 1 passing through an NLFC component 2, where the plane of the diagram is a plane of the beam in which the beam is focussed. As the focussed input beam 1 propagates through the NLFC component 2, the beam width 3 first decreases, then reaches a minimum value and then increases. The beam width 3 in a given plane of the beam is the width along a direction which lies in said plane and is perpendicular to the propagation direction 4 of the beam. The beam radius is defined as equal to one half of the beam width. The position along the propagation direction of the beam at which the beam width in a plane of the beam is a minimum is described as a beam waist in said plane of the beam, and the beam radius in said plane at this position is described as the beam waist radius 5 in said plane. In general a light beam may not have circular symmetry about its propagation direction. In this case, at any position along the propagation direction of the beam, the beam width measured in two different planes of the beam may not be equal.
The input beam may be focussed, for example, using one or more lenses, so that the beam converges in at least one plane of the beam as it propagates towards the NLFC component (that is, the width of the beam in at least one plane of the beam decreases as the beam propagates towards the NFLC component) such that a beam waist in at least one plane of the beam forms in an NLFC component. In this case, the strength of focusing of the input beam, in a given plane of the beam, can be quantified by the convergence half-angle of the light beam as measured in air before the light beam enters the NLFC component. This is also illustrated in FIG. 1. The convergence half-angle of the beam, φc, is defined as tan(φc)=½{w(z1)−w(z2)}/{z2−z1}, where w(z1) and w(z2) are widths of the beam measured at two positions along the propagation direction 4 of the beam, z1 and z2, both located before the beam propagates into the NLFC component 2.
The strength of focusing of an input light beam, in a given plane of the beam, can also be quantified by the divergence half-angle of the light beam as measured in air after the beam has exited from the NLFC component. The divergence half-angle of the beam, φd, is defined as tan (φd)=½{w(z4)−w(z3)}/{z4−z3}, where w(z3) and w(z4) are the widths of the beam measured at two positions along the propagation direction 4 of the beam, z3 and z4, both located after the beam propagates out of the NLFC component 2.
The efficiency of NLFC can depend strongly on the widths of the input beam as it propagates through the NLFC component. Boyd and Kleinman [Journal of Applied Physics 39, 3597 (1968)] describes a method to identify suitable widths for the input beam within the NLFC component to obtain SHG with high efficiency for the special case of an input beam which has the form of a circular “Gaussian beam” inside the NLFC component. A Gaussian beam is a beam of light in which the intensity of the light varies in a Gaussian way with radial distance from its centre. The width of a Gaussian beam at a position along the propagation direction of the beam is commonly defined as the distance, measured along a direction which is perpendicular to the propagation direction of the beam, between the two points at which the intensity of the beam is equal to 1/e2 of the maximum intensity of the beam at that position along the propagation direction of the beam. A circular Gaussian beam has an equal beam width in all planes of the beam, at a particular position along the propagation direction of the beam. For a Gaussian beam with wavelength λ (measured in a vacuum), propagating in a medium with refractive index n, the beam waist radius rw and the far-field convergence or divergence half-angle φ are related by equation 1.
                                          r            w                    ⁢          ϕ                =                  λ                      n            ⁢                                                  ⁢            π                                              Equation        ⁢                                  ⁢        1            
The beam radius, r, varies according to equation 2 where z is a distance from the position of the beam waist, measured along the propagation direction of the beam.
                              r          ⁡                      (            z            )                          =                              r            w                    ⁢                                    1              +                                                (                                                            λ                      ⁢                                                                                          ⁢                      z                                                              n                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                                              r                        w                        2                                                                              )                                2                                                                        Equation        ⁢                                  ⁢        2            
The method described by Boyd and Kleinman applies to NLFC in NLFC components which may exhibit “walkoff” of either the input beam or the generated beam. Walkoff can occur due to an optical birefringence of the material in the NLFC component. For the specific case of SHG, if the generated beam exhibits walkoff then the propagation direction of the generated beam in the NLFC component is not the same as the propagation direction of the input beam in the NLFC component. The walkoff angle, ρ, is defined as the angle between the propagation directions of the input beam and the generated beam. The plane containing both the input beam propagation direction and the generated beam propagation direction is referred to as the walkoff plane. The walkoff angle, ρ, can be measured experimentally. Alternatively, the walkoff angle, ρ, can be calculated using standard methods, from knowledge of the propagation direction of a beam, the orientation of the polarisation of the beam and the refractive indices of the material in the NLFC component.
For the special case of a circular Gaussian beam, the method described by Boyd and Kleinman can be applied to determine the suitable beam waist radius for the input beam inside an NLFC component to obtain high SHG efficiency. The suitable beam waist radius depends on the wavelengths of the input and generated light, the refractive indices of the NLFC component to the input and generated light, the walkoff angle and the length of the NLFC component. The refractive indices of the NLFC to the input and generated light and the walkoff angle may depend on the propagation direction of the input light within the NLFC component, relative to the optical axis, or optical axes, of the NLFC component. The convergence half-angle in air of the input beam which will provide approximately said beam waist radius within the NLFC component can then be identified using standard calculation methods for Gaussian beam optics such as using ray transfer matrices, which are sometimes referred to as “ABCD matrices”.
Freegarde et al. [Journal of the Optical Society of America B 14, 2010 (1997)] describes a method to identify suitable widths for the input light beam within the NLFC component to obtain SHG with high efficiency for the special case of an input light beam which has the form of an elliptical Gaussian beam. An elliptical Gaussian beam is a beam of light in which the intensity of the light varies in a Gaussian way with distance from the propagation axis along two directions which are perpendicular to each other and perpendicular to the propagation direction of the beam.
The method described by Freegarde et al. applies to NLFC in NLFC components which may exhibit walkoff of either the input beam or the generated beam. For the special case of an elliptical Gaussian beam, the method described by Freegarde et al. can be applied to determine the suitable beam waist radii, defined along two perpendicular principal axes of the Gaussian beam for the input beam inside the NLFC component to obtain high SHG efficiency.
The methods described by Boyd and Kleinman and by Freegarde et al. apply to “ideal” Gaussian beams. Sometimes these ideal Gaussian beams are referred to as “diffraction-limited” beams. In most devices which use SHG the input beam is an ideal Gaussian beam or is closely approximated by a Gaussian beam. In these cases the strength of focusing of input light identified by the methods described by Boyd and Kleinman or by Freegarde et al. as being suitable to obtain SHG with high efficiency are appropriate.
Some beams of light are not closely approximated by a Gaussian beam. These non-Gaussian beams are sometimes described as “non-diffraction-limited”. The beam quality factor, M2, can be used to describe the properties of non-diffraction-limited beams. For a non-diffraction-limited beams with wavelength λ (measured in a vacuum), propagating in a medium with refractive index n, the beam waist radius rw in a plane of the beam and the far-field convergence or divergence angle φ in the same plane of the beam are related by equation 3.
                                          r            w                    ⁢          ϕ                =                                            M              2                        ⁢            λ                                n            ⁢                                                  ⁢            π                                              Equation        ⁢                                  ⁢        3            
The beam radius in the plane of the beam, r, varies according to equation 4, where z is a distance from the position of the beam waist in that plane of the beam, measured along the direction of propagation of the beam.
                              r          ⁡                      (            z            )                          =                              r            w                    ⁢                                    1              +                                                (                                                                                    M                        2                                            ⁢                      λ                      ⁢                                                                                          ⁢                      z                                                              n                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                                              r                        w                        2                                                                              )                                2                                                                        Equation        ⁢                                  ⁢        4            
A non-diffraction-limited beam may be described as an elliptical beam. In this description, two planes of the beam which are perpendicular to one another are defined as principal planes of the beam. The beam quality factor, beam waist radius and the position of the beam waist measured along the propagation direction of the beam, may be different in the two principal planes of the beam.
A light beam with larger beam quality factors is generally considered to be further from diffraction-limited (i.e. “lower quality”) than a light beam with smaller beam quality factors. A light beam with M2=1 behaves as a diffraction-limited Gaussian beam; a light beam with M2>1 is a non-diffraction-limited beam.
The width of a non-diffraction-limited beam may be determined using the so called d4σ (d 4 sigma) method, as described in ISO11146 (2005). The radius of a non-diffraction-limited beam is equal to one half of the width determined by the d4σ method. The beam quality factor in a plane of a given light beam may be determined by measuring the beam radius in said plane at several positions along the direction of propagation of the beam and identifying the M2 value which yields the best fit to equation 4. Procedures for accurate determination of the beam quality factor are described in ISO11146 (2005).
In German patent application DE102007063492B4 (published Apr. 15, 2010), a device for frequency-doubling a non-diffraction-limited beam is described. In this device the “focusing convergence” of the input beam is scaled down from the value that would be found using the method described by Boyd and Kleinman and would apply to a diffraction-limited beam. In particular, the formula shown in equation 5 is described.
                              θ          opt                =                                            θ              koh                        ⁢            BK                                1            +                                          (                                                      M                    2                                    -                  1                                )                            ⁢              z                                                          Equation        ⁢                                  ⁢        5            
In equation 5, θopt is the convergence half-angle of the input beam in the frequency-doubling component which is suitable for the non-diffraction-limited beam with beam quality factor M2, θkoh BK is the convergence half-angle in the frequency-doubling component which is suitable for a diffraction-limited beam and is determined using the method described by Boyd and Kleinman, and Z is a parameter taking a value between 0.2 and 5. In equation 5 we have used the symbols used in the original reference and these are not necessarily consistent with symbols used in the remainder of this application.
It can be seen from the form of equation 5 that the suitable convergence half-angle for a beam with a large beam quality factor (M2) will be significantly smaller than the convergence half-angle which is suitable for a diffraction-limited beam, as determined using the method described by Boyd and Kleinman.
Blume et al. [Proceedings of SPIE 6875 68751C (2008)] describes an alternative device for frequency-doubling a non-diffraction-limited beam. In this device it was shown by experiment that the most suitable conditions for frequency-doubling of a non-diffraction-limited input beam corresponds to focusing of the input beam so that the beam waist radius formed in the frequency-doubling component is approximately M2 times larger than the beam waist radius which would be found to be suitable for a diffraction-limited input beam using the method described by Boyd and Kleinman. By rearranging equations 1 and 3 it is clear that this is equivalent to the condition that the most suitable convergence half-angle for a non-diffraction-limited input beam, measured in the frequency-doubling component, is equal to the most suitable convergence half-angle for a diffraction-limited input beam, measured in the frequency-doubling component, as found using the method described by Boyd and Kleinman.
One application of NLFC is to generate light with wavelengths which are difficult or impossible to generate by other methods. One important example is to generate ultraviolet light using SHG. Ultraviolet light has a wavelength shorter than 400 nm.
Nishimura et al. [Japanese Journal of Applied Physics 42 5079 (2003)] describes a system to frequency-double an input beam with wavelength 418 nm, emitted from a laser diode, to obtain a generated beam with wavelength 209 nm, using a β-BaB2O4 crystal as the NLFC component. The system uses a bulky and complex optical resonator structure but the power of the generated beam is limited to a low value of 0.009 mW. A beam waist radius in the NLFC component was chosen to match the value shown to be suitable by the method described by Boyd and Kleinman, as appropriate for a circular Gaussian beam. For this specific system, the input beam waist radius was approximately 16 μm. Using standard calculation methods for Gaussian beam optics, such as using ray transfer matrices, one can show that this corresponds to a convergence half-angle in air of 0.48°.
A similar optical resonator design is disclosed in Mauda, U.S. Pat. No. 7,110,426 (issued Sep. 19, 2006) to generate ultraviolet light. A suitable beam waist radius was determined using the method described by Boyd and Kleinman.
Tangtronbenchasil et al. [Japanese Journal of Applied Physics 47 2137 (2008)] describes a system to frequency-double a laser beam with wavelength approximately 440 nm, emitted from a laser diode, to obtain a generated beam with wavelength approximately 220 nm, using a β-BaB2O4 crystal as the NLFC component. The power of the generated beam was limited to a low value of approximately 0.0002 mW. For this system the beam waist radius in the NLFC component was chosen to match the value shown to be suitable by the method described by Boyd and Kleinman, as appropriate for circular Gaussian beam.